Who am I?

I’m Professor Andrew Pontzen, a cosmologist at the Institute for Computational Cosmology, part of the physics department here at Durham. I only just started working here this academic year, so I’m still learning my way around the Durham system like you are.

Previously I was a professor at University College London, and before that I worked at Oxford and Cambridge universities. You can read more about me and my research about galaxies, dark matter, and the universe on my personal website.

Picture of THE UNIVERSE IN A BOX paperback jacket I also wrote a book recently, which you might enjoy! It’s a non-technical introduction to computer simulations of our universe. I’ll be illustrating some bits of this course with computer simulations – it helps us understand physics in ways that a static piece of paper can’t. But, of course, being able to do the necessary bits of maths on a piece of paper is also important. The combination of computer simulation and pencil-and-paper theory is an incredibly powerful way of understanding physics. I’ll give you a bit of a taste of that in this course, using some simulations embedded in your browser.

In case you are worried about what to call me, I am happy with either of:

  • Andrew
  • Professor Pontzen

Address me whichever of these ways that makes you feel more comfortable. Please avoid calling me ‘Sir’ which makes us all feel like we’re at school.

Why study electromagnetism?

Imagine a world where electromagnetism hadn’t been discovered. How would that world compare to ours?

  • Obviously, there’d be no computers, phones, internet, televisions, radios, GPS, modern transportation and so on.
  • Globalisation, built on near-instantaneous electromagnetic communication, would be essentially impossible.
  • Few if any of the major developments in 20th century music, theatre and other cultural areas would have taken place (think about the incredible influence of amplification and the electric guitar, electric lighting and more).
  • Medical scanning techniques like MRI, X-rays, CT scans, EEGs and lab equipment like electron microscopes or spectrophotometers also wouldn’t exist, severely restricting medical progress.
  • It wouldn’t have been possible to discover 20th-century theories like relativity, quantum mechanics and quantum field theory. Electromagnetism is very different to what came before it, and acts like a ‘bridge’ between Newtonian theories and modern conceptions of physics.

Whether or not you like all of the developments on this list, it’s dizzyingly difficult to imagine the alternative world where electromagnetism was never discovered. It is a keystone of modern engineering, and a conceptual leap that opens up whole new ways of thinking about physics.

People (especially in the media or government) sometimes ask what the value of research into the frontiers of physics is, and we are regularly asked to justify our activities in economic terms when they are really just about gaining knowledge. In the early days of studying electromagnetism, in the 18th and 19th centuries, studying it seemed like an esoteric curiosity. Now the entire world economy is founded upon electromagnetism, and over the coming few years it is our only hope of escaping fossil fuel dependence. With this example in mind, it is impossible to say what avenues today’s abstract physics research might open up in 100 or 200 years from now.

Discussion point: Now imagine a universe where electromagnetism not only hasn’t been discovered, but actually it doesn’t exist at all, i.e. there are no electrical or magnetic forces. What would be different?

Help!

There is a lot of information coming your way.

  • These notes are a summary of what will get covered in lectures; you can read them in advance or afterwards depending on what most helps you.
  • During lectures, we will draw diagrams and cover worked examples, in a way that complements these notes.
  • The notes point you to everything that matters but they don’t give all the detail. That’s covered in the Young & Freedman textbook, and each lecture points you to the sections that you should read.

Electromagnetism is a deep topic, and therefore it can be hard. Do not worry if you find it tricky, but do persist: read the textbook, read the notes, attend the lectures, spend some of your own time thinking, and then seek out help. You have lots of options, but only you can actually take the first step. You could:

  • talk to your peers;
  • ask in your tutorial groups;
  • drop in to office hours (see Blackboard for dates/times);
  • email me;
  • and, yes, AI can answer a lot of your electromagnetism questions accurately but beware! It makes weird mistakes. Cross-check anything it tells you very carefully.

Try out different ways of getting help and see what works for you.

Notation

Electromagnetism is all about vectors, i.e. quantities with a magnitude and direction.

There are different ways to write a vector.

  • $\mathbf{r}$: sometimes they are written in bold, e.g. in Young and Freedman. I will use this in these notes, most of the time. But don’t try to do this by hand.
  • $\overrightarrow{r}$: sometimes they are written with an arrow over the top.
  • $\underline{r}$: sometimes they are written with an underline. Normally for hand-writing, this is clearest and easiest. I will use it in lectures. It is also used in Durham physics exams.

Other important bits of notation:

  • $\vec{r}_x$, $\vec{r}_y$, $\vec{r}_z$ are the $x$, $y$ and $z$ components of the vector $\vec{r}$.
  • $\vert\vec{r}\vert$ is the magnitude (length) of the vector $\vec{r}$.
  • $\hat{\vec{r}}$ is a unit vector parallel to $\vec{r}$.
  • When considering two separate points 1 and 2, their locations are described by the vectors $\vec{r}_1$ and $\vec{r}_2$.
  • \(\vec{r}_{1,2}\) is a vector describing the displacement from point 1 to point 2; therefore $\vec{r}_{1,2} = \vec{r}_2 - \vec{r}_1$.

Vectors can be written out in components, for example:

\[\vec{r} = \begin{bmatrix} 5 \\ 2 \\ 9 \end{bmatrix}\]

But, the vector above can also be written on a single line as $\vec{r} = 5 \ihat + 2 \jhat + 9 \khat$, where $\ihat$, $\jhat$ and $\khat$ are unit vectors in the $x$, $y$ and $z$ directions:

\[\ihat = \begin{bmatrix} 1 \\ 0 \\ 0 \end{bmatrix}, \quad \jhat = \begin{bmatrix} 0 \\ 1 \\ 0 \end{bmatrix}, \quad \textrm{and}\quad \khat = \begin{bmatrix} 0 \\ 0 \\ 1 \end{bmatrix}.\]

Coulomb’s law

To see the use of vector notation, let’s immediately write down Coulomb’s law, which is the law of attraction (or repulsion) between two point charges.

When you saw this in school, it probably looked like this:

\[F = \frac{1}{4\pi\epsilon_0} \frac{q_1 q_2}{r^2}\]

Here $\epsilon_0$ is a constant of proportionality (and the subscript $0$ is supposed to indicate that this is for the electric field in a vacuum). Now, we can write it more precisely:

\[\vec{F}_{1,2} = \frac{1}{4\pi\epsilon_0} \frac{q_1 q_2}{\vert \vec{r}_{1,2} \vert^2} \hat{\vec{r}}_{1,2}\]

In the lecture, we will draw relevant diagrams and make clear what all these terms mean.

Electric fields

Coulomb’s law says there is a force between two charged objects. But this isn’t the most fundamental way to understand what is going on. In a modern view, we imagine instead that there is an “electric field” generated by the objects.

Instead of directly talking about the force, we say there is an electric field generated by the first object. If the first object is placed at the origin, the electric field at a position $\vec{r}$ is given by

\[\vec{E}(\vec{r}) = \frac{1}{4\pi\epsilon_0} \frac{q}{\vert \vec{r} \vert^2} \hat{\vec{r}}\]

and the force on a second object with charge $q_2$ is equal to

\[\vec{F} = \vec{E}(\vec{r}_2) q_2.\]
  • Substitute the equation for $\vec{E}$ into the equation for $\vec{F}$ and you will just regenerate Coulomb’s law.
  • $\vec{E}$ is a vector, i.e. it has magnitude and direction.
  • But it is also a function of $\vec{r}$, i.e. it takes different magnitudes and directions in different places.
  • This is the definition of a vector field — a vector which depends on position.
  • We can draw a vector field by having a grid of arrows in space — this is called a ‘quiver’ plot (or occasionally a ‘fletchings’ plot).
  • To make it clear we’ll need to stick to 2D rather than try to show it in full 3D.

Here’s what that looks like. You can try dragging the charge around to see how the electric field responds.

Discussion point: In what sense is the electric field ‘real’? In the example above, the field seems like a purely optional way of accounting for the existence of a force. Why do physicists see fields as fundamental, and non-optional? Are ‘gravitational fields’ real things, or just electric/magnetic fields?

Fieldline visualization

There is another way of visualising a field that we will sometimes use; it is called a fieldline visualization. Here is the same charge from above, but now using fieldlines. Again, you can drag it around to see how the fieldlines follow the charge.

  • These kind of plots are much easier to sketch than the fletchings, and clearly show the direction of the field although the magnitude of the field is a bit less clear.
  • The most important property is that the fieldlines are everywhere tangent to the field; in a more intuitive sense, the lines ‘follow the direction’ of the field.
  • The density of the fieldlines roughly1 indicates the strength of the field $\vert \vec{E}\vert$.

Sometimes you will end up sketching fields, and you should always use fieldlines in this case (it’s too hard to try and make quiver plots by hand). There are some rules to follow to make a clear fieldline plot:

  1. Fieldlines start on positive charges (or at the edge of the plot) and end on negative charges (or, again, at the edge of the plot).
  2. Fieldlines approach/depart a charge radially, and are as close to equally spaced as possible.
  3. Fieldlines always follow the direction of the field at each point they are passing through.
  4. Fieldlines do not cross each other. (Can you see why not?)

Footnotes

I’ve added footnotes to a few lectures where there is some additional information that you might be interested in. Footnotes are never examinable.

  1. Why does the density of drawn fieldlines roughly follow the strength of the field? According to Coulomb’s law, as you go further away from a charge where the fieldlines originate, the strength of the field declines (as $r^{-2}$, where $r$ is the distance from the source). If the fieldlines could be drawn in 3D space (rather than on a 2D page) the density of fieldlines would also decline as $r^{-2}$. But we are drawing them instead in 2D space, which means the density of lines actually declines as $r^{-1}$. So, roughly, we see the right behaviour but mathematically it is only correct for a 2D universe rather than the 3D one we inhabit. ↩